Bridging the gap between mechanistic and adaptive explanations of territory formation

Abstract

How animals divide space can have fundamental implications for the population dynamics of territorial species. It has recently been proposed that space can be divided if animals tend to avoid fight locations, rather than the winner of fights gaining access to exclusive resources, behaviour that generates exclusive territories in two-dimensional space. A game-theory model has shown that this avoidance behaviour can be adaptive, but the adaptiveness has not been investigated in a spatially realistic context. We present a model that investigates potential strategies for the acquisition of territories when two-dimensional space must be divided between individuals. We examine whether exclusive territories form when animals avoid all encounters with others, or only those encounters that have led to losing fights, under different fighting costs and population densities. Our model suggests that when fighting costs are high, and the population density is low, the most adaptive behaviour is to avoid fight locations, which generates well-defined, exclusive territories in a population that is able to resist invasion by more aggressive strategies. Low fighting costs and high population densities lead to the break-down of territoriality and the formation of large, overlapping home ranges. We also provide a novel reason as to why so-called paradoxical strategies do not exist in nature: if we define a paradoxical strategy as an exact mirror-image of a common-sense one, it must respond in the opposite way to a draw as well as to wins and losses. When this is the case, and draws are common (fight outcomes are often not clear-cut in nature), the common-sense strategy is more often adaptive than a paradoxical alternative.

Appendix: model of spatial division

The following steps outline the simulation model of spatial division.

• 1. Define the strategies and other parameters used in the model. Select the two strategies S₁ and S₂ to be tested against each other.

• 2. n individuals are each allocated one of the two selected strategies. The number of individuals allocated each strategy is determined by the parameter f. For a single invading mutant, f=1/n. The first fn individuals are allocated strategy S₁, and the remainder, strategy S₂

• 3. The n individuals arrive in a grid of squares measuring a squares by a squares. The grid of squares is wrapped such that each square has exactly four neighbouring squares. The initial location of each individual (i, j) is determined randomly, and is independent of the location of all other individuals.

• 4. A for each individual (k) is set to 1 in the initial location square (i, j), such that A(i, j, k, t₁)=1. A in all other squares is set to zero.

• 5. For each individual, A in all squares surrounding its initial location increase by εΑ (which in this case is equal to ε).

• 6. Each individual uses all squares in which A(i, j, k, t)>0. For each individual, k₁ to k_n, we compare the location identities of all squares in which A(i, j, k, t)>0 with the location identities of all other individuals.

• 7. For each possible pair of individuals, we record the location where both A(i, j, k₁, t)>0 and A(i, j, k₂, t)>0. At this location, a fight takes place.

• 8. The outcome of the fight is determined randomly, with a probability d that the fight ends in a draw. If a number drawn from a random number distribution is less than (1-d)/2, we record that individual k₁ won the fight, and individual k₂ lost. If the random number is between (1-d)/2 and d, we record a loss for individual k₁ and a win for individual k₂. Otherwise, we record a draw for both individuals. This step is repeated for all possible pairs of individuals in all locations where both A(i, j, k₁, t)>0 and A(i, j, k₂, t)>0.

• 9. For each fight, the costs to the each of the participants is recorded, and added to any existing costs already paid by the individual from other fights in the same or previous time-steps.

• 10. As a result of the outcome of fights, A(i, j, k, t) changes in accordance with the individual strategy, by a value δ. For each fight recorded above, we determine how A will change as a result of the value of δ and the strategy used by the individuals involved in that fight. These changes (some of which are positive and some negative) are recorded for each square for each individual.

• 11. If there is no fight in a particular square for a particular individual, A changes in accordance to their response to finding the space empty. Combining steps 10 and 11 gives a change in A for each square, for each individual.

• 12. When all the changes as a result of the fights have been recorded, they are added to the original A value for each square. Any A values which then exceed 1 or are below zero are set to 1 and 0 respectively.

• 13. At each location (i, j) in which A(i, j, k, t)>0, the surrounding squares (i–1, j), (i+1, j), (i, j–1), (i, j+1) all increase their value of A by the amount εA(i, j).

• 14. Steps 6–13 are repeated for t_max times. After this, the simulation ends.

• 15. At the end of the simulation, individual fitness, home range size and number of squares used exclusively are calculated, and grouped according to the strategy used by the individual. Strategy means are then calculated and collected.

• 16. The entire simulation is repeated 20 times.